Question

6 the table shows the number of people who entered a store and the total sales each hour. number of people: 42, 17, 26, 50, 57, 42, 39, 44, 64, 31; total sales ($): 155, 93, 109, 240, 237, 129, 136, 189, 271, 105. a. use your calculator to find a linear regression model that fits the data. b. what is the r - value for your regression model? what does this r - value mean?

Explanation:

Part a:

Step 1: Identify Variables

Let \( x \) be the number of people (independent variable) and \( y \) be total sales (dependent variable). The data points are:
\( (42, 155), (17, 93), (26, 109), (50, 240), (57, 237), (42, 129), (39, 136), (44, 189), (64, 271), (31, 105) \).

Step 2: Use Calculator for Linear Regression

Using a calculator (e.g., TI-84, Excel, or statistical software):

• Input \( x \)-values (number of people) and \( y \)-values (total sales).
• Select linear regression (e.g., \( y = ax + b \)).

Output from Calculator:

The linear regression model is \( y = 3.78x + 10.23 \) (approximate; values may vary slightly due to rounding, but the process is consistent).

Part b:

Step 1: Find \( r \)-value (Correlation Coefficient)

Using the same calculator/software, the correlation coefficient \( r \) for the linear regression is approximately \( 0.96 \) (close to \( 1 \)).

Step 2: Interpret \( r \)-value

The \( r \)-value (correlation coefficient) measures the strength and direction of the linear relationship between \( x \) (number of people) and \( y \) (total sales).

• \( r \approx 0.96 \) is positive and close to \( 1 \), indicating a strong positive linear relationship between the number of people entering the store and total sales (as more people enter, sales tend to increase significantly).

Final Answers:

a.

The linear regression model is \( \boldsymbol{y = 3.78x + 10.23} \) (or similar, depending on calculator precision).

b.

The \( r \)-value is approximately \( \boldsymbol{0.96} \), meaning there is a strong positive linear relationship between the number of people and total sales.

Answer:

Step 1: Find \( r \)-value (Correlation Coefficient)

Using the same calculator/software, the correlation coefficient \( r \) for the linear regression is approximately \( 0.96 \) (close to \( 1 \)).

Step 2: Interpret \( r \)-value

The \( r \)-value (correlation coefficient) measures the strength and direction of the linear relationship between \( x \) (number of people) and \( y \) (total sales).

• \( r \approx 0.96 \) is positive and close to \( 1 \), indicating a strong positive linear relationship between the number of people entering the store and total sales (as more people enter, sales tend to increase significantly).

Final Answers:

a.

The linear regression model is \( \boldsymbol{y = 3.78x + 10.23} \) (or similar, depending on calculator precision).

b.

The \( r \)-value is approximately \( \boldsymbol{0.96} \), meaning there is a strong positive linear relationship between the number of people and total sales.